Poisson Regression | Mplus Data Analysis Examples

Version info: Code for this page was tested in Mplus version 6.12.

Poisson regression is used to model dependent variables that are counts.

Please note: The purpose of this page is to show how to use various data
analysis commands. It does not cover all aspects of the research process which
researchers are expected to do. In particular, it does not cover data
cleaning and checking, verification of assumptions, model diagnostics or
potential follow-up analyses.

Examples of Poisson regression

Example 1. The number of persons killed by mule or horse kicks in the
Prussian army per year. von Bortkiewicz collected data from 20 volumes of
Preussischen Statistik. These data were collected on 10 corps of
the Prussian army in the late 1800s over the course of 20 years.

Example 2. The number of people in line in front of you at the grocery store.
Predictors may include the number of items currently offered at a special
discounted price and whether a special event (e.g., a holiday, a big sporting
event) is three or fewer days away.

Example 3. The number of awards earned by students at a single high school.
Predictors of the number of awards earned include the type of program in which the
student was enrolled (e.g., vocational, general or academic) and the score on their
final exam in math.

Description of the data

Let’s pursue Example 3 from above.

The data for this example were simulated and are in the file
https://stats.idre.ucla.edu/wp-content/uploads/2016/02/poisson_sim.dat.
In this example, num_awards is the outcome variable and indicates the
number of awards earned by students at a single high school in a single year, math is a continuous
predictor variable and represents students’ scores on their math final exam, and prog is a categorical predictor variable with
three levels indicating the type of program in which the students were
enrolled.

Let’s look at the data. It is always a good idea to start with descriptive
statistics.

Analysis methods you might consider

Below is a list of some analysis methods you may have
encountered. Some of the methods listed are quite reasonable, while others have
either fallen out of favor or have limitations.

Poisson regression – Poisson regression is often used for modeling count
data. Poisson regression has a number of extensions useful for count models.

Negative binomial regression – Negative binomial regression can be used for over-dispersed
count data, that is when the conditional variance exceeds the conditional
mean. It can be considered as a generalization of Poisson regression since
it has the same mean structure as Poisson regression and it has an extra
parameter to model the over-dispersion. If the conditional
distribution of the outcome variable is over-dispersed, the confidence intervals for
Negative binomial
regression are likely to be narrower as compared to those from a Poisson regression.

Zero-inflated regression model – Zero-inflated models attempt to account
for excess zeros. In other words, two kinds of zeros are thought to
exist in the data, "true zeros" and "excess zeros". Zero-inflated
models estimate two equations simultaneously, one for the count model and one for the
excess zeros.

OLS regression – Count outcome variables are sometimes log-transformed
and analyzed using OLS regression. Many issues arise with this
approach, including loss of data due to undefined values generated by taking
the log of zero (which is undefined) and biased estimates.

Poisson regression analysis

In the Mplus syntax below, we specify that the variables to be used in the
Poisson regression are num_awards, p2, p3 and math.
(The variables p2 and p3 are indicator variables for prog.) We also specify that num_awards is a count variable. (Because the
variable name num_awards has more than eight characters, we get a warning in the
output that this variable name has been truncated to eight characters.) By
default, Mplus uses restricted maximum likelihood (MLR), so robust standard
errors are given in the output. The MLR standard errors are computed using
a sandwich estimator. These are what we generally call robust standard
errors. Cameron and Trivedi (2009) recommend the use
of robust standard errors when estimating a Poisson model. If you do not want robust standard errors, you can use the
analysis: estimator = ml; block.

After the heading informing that "THE MODEL ESTIMATION TERMINATED NORMALLY"
comes the information about the model. It begins with the information
regarding the log
likelihood, AIC and BIC. These values can be used in comparing models.

Then we find the
Poisson regression coefficients for each of the variables along with the
(robust) standard
errors. The column labeled as Est./S.E. is the
quotient of the estimates divided by the standard errors. These are basically
z-scores if the sample size is reasonably large. In the right-most column
is the two-tailed p-value.

The indicator variable
p2 is
statistically significant. Compared to level 1 of prog, the
expected log rate for level 2 of prog increases by about 1.1. The
indicator variable p3 is not statistically significant. The coefficient for math
is .07 and is statistically significant. This means that the expected
log count increase in num_awards for a one-unit increase in math is .07.

To determine if prog itself is statistically significant, we can
use the model test block to obtain the
two degree-of-freedom test of this variable.

In the syntax below, some of the variables in the model are given labels. These labels must be in parentheses and must be
the last item listed on the line, so the model is broken up over several lines. We have given the label
a2 to the indicator
variable p2, and the label a3 to the indicator variable p3. Once we have assigned labels to the variables, we can use those
labels in the model test block. Setting both a2 and a3 to 0 allows us to get the two degree-of-freedom test of the variable
prog.

We can see that the variable prog, as a whole, is statistically significant.
To help assess the fit of the model, we can look at the model fit statistics in the output. Several measures of goodness of fit
are provided. For both the AIC and BIC, smaller is better.

To obtain the results as incident rate ratios, we need to use the model
constraint block. Again, we use labels to refer to the variables
in the model. In the model constraint block, we use the new
statement to label the new parameters, which will be the exponentiated
parameters from the model.

Things to consider

When there seems to be an issue of dispersion, we should first check if
our model is appropriately specified, such as omitted variables and
functional forms. For example, if we omitted the predictor variable prog
in the example above, our model would seem to have a problem with
over-dispersion. In other words, a mis-specified model could present a
symptom like an over-dispersion problem.

Assuming that the model is correctly specified, you may want to test for
over-dispersion. There are several tests of the alpha parameter,
including the likelihood ratio test.

One common cause of over-dispersion is excess zeros, which in turn are
generated by an additional data generating process. In this situation,
zero-inflated model should be considered.

If the data generating process does not allow for any 0s (such as the
number of days spent in the hospital), then a zero-truncated model may be
more appropriate.

Count data often have an exposure variable, which indicates the number
of times the event could have happened. This variable should be
incorporated into your Poisson model by taking the log of the
exposure variable and constraining its estimate to 1.

The outcome variable in a Poisson regression cannot have negative numbers, and the exposure
cannot have 0s.

The diagnostics for Poisson regression are different from those for OLS
regression. The assumptions of the model should be checked (see
Cameron and Trivedi (1998) and Dupont (2002) for more information).

Poisson regression is estimated via maximum likelihood estimation. It
usually requires a large sample size.